Existence results and bifurcation for nonlocal fractional problems with critical Sobolev exponent

This paper is concerned with existence and bifurcation of nontrivial solutions for the following critical nonlocal problem −LKu=λf(x,u)+|u|2∗−2uinΩ,u=0inRn∖Ω,where α∈(0,1), Ω is an open bounded subset of Rn(n>2α) with continuous boundary, λ is a positive real parameter, 2∗=2nn−2α is the fractiona...

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Bibliographic Details
Published in:Computers & mathematics with applications (1987) 2019-09, Vol.78 (5), p.1720-1731
Main Authors: Li, Pan-Li, Sun, Hong-Rui
Format: Article
Language:English
Subjects:
Bifurcation
Bifurcation theory
Critical Sobolev exponent
Eigenvalue
Elliptic functions
Nonlocal
Operators (mathematics)
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Summary:This paper is concerned with existence and bifurcation of nontrivial solutions for the following critical nonlocal problem −LKu=λf(x,u)+|u|2∗−2uinΩ,u=0inRn∖Ω,where α∈(0,1), Ω is an open bounded subset of Rn(n>2α) with continuous boundary, λ is a positive real parameter, 2∗=2nn−2α is the fractional critical Sobolev exponent and f satisfies suitable growth condition, LK is the integrodifferential operator defined as LKu(x)=∫Rn(u(x+y)+u(x−y)−2u(x))K(y)dy,x∈Rn.The single solution results extend the main results of Servadei et al. (2015) [3,18], and include Theorem 1.1 of Servadei (2013), the bifurcation result extends those got by Sang (1994) for classical elliptic equations, to the case of nonlocal fractional operators, and also generalizes the result got by Fiscella et al. (2016).
ISSN:0898-1221
1873-7668
DOI:10.1016/j.camwa.2019.04.005